A Hierarchy of Polyhedral Arrangements

Polyhedral Algebras, Arrangements of Toric Varieties, and Their Groups

We investigate the automorphism groups of graded algebras defined by lattice polyhedral complexes and of the corresponding projective varieties, which form arrangements of projective toric varieties. These groups are polyhedral versions of the general and projective linear groups. It is shown that for wide classes of complexes they are generated by toric actions, elementary transformations and ...

Arrangements and their Applications in Robotics : Recent

We survey a collection of recent combinatorial and al-gorithmic results in the study of arrangements of low-degree algebraic surface patches in three or higher dimensions. The new results extend known results involving 2-dimensional arrangements, and they almost settle several conjectures posed eight years ago. Arrangements play a central role in the design and analysis of geometric algorithms,...

Polyhedral Kähler Manifolds

In this article we introduce the notion of Polyhedral Kähler manifolds, even dimensional polyhedral manifolds with unitary holonomy. We concentrate on the 4-dimensional case, prove that such manifolds are smooth complex surfaces, and classify the singularities of the metric. The singularities form a divisor and the residues of the flat connection on the complement of the divisor give us a syste...

Sparse Arrangements and the Number of Views of Polyhedral Scenes

Symmetric Tensor Approximation Hierarchies for the Completely Positive Cone

In this paper we construct two approximation hierarchies for the completely positive cone based on symmetric tensors. We show that one hierarchy corresponds to dual cones of a known polyhedral approximation hierarchy for the copositive cone, and the other hierarchy corresponds to dual cones of a known semidefinite approximation hierarchy for the copositive cone. As an application, we consider a...